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A062734 Triangular array T(n,k) giving number of connected graphs with n labeled nodes and k edges (n >= 1, 0 <= k <= n(n-1)/2). 7
1, 0, 1, 0, 0, 3, 1, 0, 0, 0, 16, 15, 6, 1, 0, 0, 0, 0, 125, 222, 205, 120, 45, 10, 1, 0, 0, 0, 0, 0, 1296, 3660, 5700, 6165, 4945, 2997, 1365, 455, 105, 15, 1, 0, 0, 0, 0, 0, 0, 16807, 68295, 156555, 258125, 331506, 343140, 290745, 202755, 116175, 54257, 20349 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

T(n,n-1) = n^(n-2) counts free labeled trees A000272.

T(n,n) counts labeled connected unicyclic graphs A057500. - Geoffrey Critzer, Oct 07 2012

REFERENCES

Cowan, D. D.; Mullin, R. C.; Stanton, R. G. Counting algorithms for connected labelled graphs. Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), pp. 225-236. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man., 1975. MR0414417 (54 #2519). - N. J. A. Sloane, Apr 06 2012

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, Page 29, Exercise 1.5.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..9919 (terms 1..75 from Alex Ermolaev, terms 76..175 from Alois P. Heinz)

R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017), Table 58.

FORMULA

G.f.: Sum_{n>=1, k>=0} T(n, k) * x^n/n! * y^k = log(Sum_{n>=0} (1 + y)^binomial(n, 2) * x^n/n!). - Ralf Stephan, Jan 18 2005

EXAMPLE

Triangle starts:

[1],

[0, 1],

[0, 0, 3,  1],

[0, 0, 0, 16,  15,   6,   1],

[0, 0, 0,  0, 125, 222, 205, 120, 45, 10, 1],

...

MATHEMATICA

nn=6; s=Sum[(1+y)^Binomial[n, 2] x^n/n!, {n, 0, nn}]; Range[0, nn]!CoefficientList[Series[Log[ s]+1, {x, 0, nn}], {x, y}]//Grid  (* returns triangle indexed at n = 0, Geoffrey Critzer, Oct 07 2012 *)

T[ n_, k_] := If[ n < 0, 0, Coefficient[ n! SeriesCoefficient[ Log[ Sum[ (1 + y)^Binomial[m, 2] x^m/m!, {m, 0, n}]], {x, 0, n}], y, k]]; (* Michael Somos, Aug 12 2017 *)

PROG

(PARI) {T(n, k) = if( n<0, 0, n! * polcoeff( polcoeff( log( sum(m=0, n, (1 + y)^(m * (m-1)/2) * x^m/m!)), n), k))}; /* Michael Somos, Aug 12 2017 */

CROSSREFS

Cf. (row sums) A001187, (unlabeled case) A054924, (a subdiagonal) A061540.

See A123527 for another version (without leading zeros in each row).

Sequence in context: A300812 A144209 A094544 * A205531 A269246 A247505

Adjacent sequences:  A062731 A062732 A062733 * A062735 A062736 A062737

KEYWORD

easy,nonn,tabf

AUTHOR

Vladeta Jovovic, Jul 12 2001

STATUS

approved

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Last modified July 21 22:02 EDT 2019. Contains 325210 sequences. (Running on oeis4.)